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Sample Size Calculator

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Asking everybody is expensive and usually impossible, so surveys ask a sample and accept a known amount of uncertainty in exchange. This works out how large that sample has to be for the precision you want — or, going the other way, how precise the answer you already have actually is. The z-score is computed rather than looked up in a table of four values, so any confidence level works, and a finite population correction is applied whenever you know how many people there are in total.

What do you need?
%

±5 is common; ±3 needs roughly three times the sample

z = 1.9600

%

Leave at 50 unless you have real prior evidence — it needs the largest sample, so it is the safe assumption

Leave blank if very large. Below a few thousand it makes a real difference.

%

Adds how many invitations you would need to send

How to use the sample size calculator

  1. 1Choose whether you are planning a survey or assessing one you have already run.
  2. 2Set the margin of error you can live with — ±5% is the usual starting point.
  3. 3Choose a confidence level. 95% is the convention and means the interval would contain the true value in 95 of 100 repeated samples.
  4. 4Leave the response distribution at 50% unless you have genuine prior evidence about the split.
  5. 5Enter the population size if it is small enough to matter — below a few thousand it makes a real difference.
  6. 6Add an expected response rate to see how many invitations that implies sending.

Examples

The default national poll

Input
±5%, 95% confidence, 50% distribution
Result
385 responses

This is why so many published polls quote around 1,000 respondents for ±3% — the sample grows with the square of the precision.

A small organisation

Input
±5%, 95% confidence, population of 500
Result
218 responses instead of 385

The finite population correction nearly halves the requirement once the sample is a large share of the whole.

Assessing a result you already have

Input
400 responses, 62% said yes, 95% confidence
Result
±4.76% — the true figure is very likely between 57.2% and 66.8%

Wide enough that a rival option at 58% would not be reliably behind.

About the sample size calculator

Where the formula comes from

When you sample repeatedly from a population and record the proportion each time, those proportions cluster around the true value in a bell curve — a consequence of the central limit theorem, which is what makes any of this possible. The spread of that curve is the standard error, and it equals the square root of p(1−p) divided by n. Multiply the standard error by a z-score chosen for your confidence level and you have the margin of error.

Rearranging for n gives the formula this page uses: z² times p(1−p), divided by the margin squared. Everything follows from there. The margin is squared, which is why precision is so expensive. The p(1−p) term peaks at a half, which is why 50% is the safe default. And n does not depend on the population size at all, which is the result that surprises people most.

The finite population correction is the one adjustment to that picture. The formula above assumes sampling with replacement from an infinite pool, and when the population is small each person you have already asked genuinely reduces what is left to learn. Multiplying by the correction accounts for it, and its effect fades quickly as the population grows.

The z-score, and why it is calculated here

The z-score is the number of standard errors you have to reach out from the centre to capture your chosen share of the distribution. For 95% confidence it is 1.95996, usually rounded to 1.96; for 99% it is 2.5758. Most calculators carry a short table of these and refuse anything not on it, which quietly limits what you can ask.

This page computes the inverse normal distribution directly, using a rational approximation refined against the complementary error function. That means any confidence level works — including the awkward ones that come up when a client asks for 97.5%, or when you are reconstructing a figure somebody else published without saying which level they used.

One consequence worth knowing: the relationship between confidence and sample size is steep at the top end. Moving from 95% to 99% confidence multiplies the required sample by about 1.7, and moving to 99.9% multiplies it by nearly three. Very high confidence is rarely worth what it costs in a survey context, which is why 95% became the convention in the first place.

Frequently asked questions

Why does the population size barely matter for large populations?
Because precision comes from the size of the sample, not from the fraction of the population it represents. A random sample of 1,000 describes a city of a million about as well as it describes a country of a hundred million — the finite population correction only bites when the sample is a substantial share of the whole, roughly above 5%. It is the reason a national poll and a city poll quote similar sample sizes, which strikes most people as wrong until the arithmetic is in front of them.
What does 95% confidence actually mean?
It describes the procedure rather than the particular answer. If you repeated the same survey many times with fresh random samples, about 95% of the intervals you calculated would contain the true population value. It does not mean there is a 95% probability that this specific interval contains it — the true value is fixed and either sits inside or does not. The distinction sounds pedantic and matters, because the looser reading invites treating one survey as more certain than it is.
Why is 50% the default response distribution?
Because it needs the largest sample, so it is the assumption that cannot leave you short. The variance term is p(1−p), which peaks at p = 0.5 and falls away towards either end — a question you expect to split 90/10 needs only about a third as many responses as one expected to split evenly. Narrowing it is legitimate when you have real prior data, and a guess dressed up as prior data buys a smaller sample and a wider true error.
Halving the margin of error more than doubles my sample. Why?
The margin is inversely proportional to the square root of the sample size, so precision improves with the square root while cost grows linearly. Going from ±5% to ±2.5% needs four times the responses; going to ±1% needs twenty-five times. This is the single most important economic fact about surveying, and it is why almost everyone settles somewhere between ±3% and ±5% rather than paying for certainty.
Does hitting this number make my survey accurate?
No — it makes the sampling error small, which is only one of the ways a survey goes wrong. If the people who responded differ systematically from those who did not, the result is biased no matter how many of them there were, and no sample size fixes that. Leading questions, an unrepresentative panel and a topic that attracts people who feel strongly are all larger sources of error than the interval on this page, and none of them appears in the arithmetic.

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